Random binary waveforms with specific correlation properties are required for ranging and other applications, especially in radar systems. It is particularly desirable to provide random binary waveforms with maximum unpredictability, hence with low probability of intercept, and also resistant to intelligent jamming. Furthermore, such random binary waveforms are also useful for applications in multi-user environments where many similar or disparate systems operate in the same geographical region and those systems share, at least partly, the same wide frequency band.
The generation of binary waveforms with specified correlation properties is of considerable practical interest in the field of radar and communications. For example, in low probability of intercept (LPI) radar the phase of the coherent carrier is modulated by a pseudo-random binary waveform to spread the spectrum of the transmitted signal. In some applications, such as collision avoidance/obstacle detection, altimetry, autonomous navigation etc., many similar radar systems should be capable of operating in the same region and sharing the same wide frequency band. To avoid mutual interference, each system should use a distinct signal, preferably orthogonal to the signals employed by all other systems. Therefore, the successful use of coded-waveform radar in a multi-user environment depends on the availability of large families of waveforms, each with specified correlation properties and low cross correlation values.
An important class of synchronous binary waveforms can be obtained from suitably constructed binary sequences, such as pseudo-random binary sequences. However, when the number and type of systems (co-operating or unco-operating) sharing the same frequency band is unknown and often cannot even be predicted, it is not possible to assign a distinct binary sequence to each of them. It is also difficult to construct large sets of long pseudo-random sequences that provide a significant improvement over purely random sequences.
The above problems can be avoided, or at least alleviated, when asynchronous random binary waveforms are used. In dense signal environments asynchronous waveforms are known to be superior to synchronous ones as a result of the additional randomisation of the zero crossing time instants. Because purely random binary waveforms exhibit maximum unpredictability, they are less vulnerable to intercept and intelligent jamming.
One convenient and inexpensive method to generate a random binary waveform is based on level crossings of a random signal generated by a physical noise source. FIG. 1 shows an example of a generator of a random binary waveform. The generator comprises a physical noise source (PNS) and a zero-crossing detector (ZCD) which can be a comparator or a hard limiter. FIG. 2 shows a typical realisation of a noise signal s(t) and a random binary waveform b(t) obtained from that noise signal and defined by zero crossings of that signal. Each zero crossing results in an event (an edge) in the binary waveform b(t), the events occurring aperiodically and unpredictably.
In radar and also other applications the shape of the correlation function of a binary waveform is of primary importance. The ideal correlation function would have the form of an impulse (Dirac delta) function. In practice, the correlation function of a ‘good’ binary waveform should attempt to approximate in some way this ideal shape. FIG. 3 shows the shape of the correlation function Rb(τ) of a random binary waveform b(t) ideal for ranging applications.
In practice it is relatively easy to generate noise signals with a Gaussian distribution, e.g., by exploiting thermal noise. When an underlying noise signal s(t) has a Gaussian distribution, the correlation function Rb(τ) of a binary waveform b(t) obtained from zero crossings of the signal s(t) can be determined from Van Vleck's formulaRb(τ)=(2/π) arcsin [Rs(τ)]where Rs(τ) is the correlation function of the underlying noise signal s(t). Therefore, in order to obtain a narrow correlation function Rb(τ) of a random binary waveform b(t), the correlation function Rs(τ) of an underlying noise signal s(t) should also be narrow. Because the correlation function and the power spectral density of a random signal form a Fourier pair, a physical noise source utilised to generate a binary waveform with a narrow correlation function should produce a noise signal with an extremely wide frequency spectrum.
It is known that the correlation function of a random binary waveform, not necessary obtained from a Gaussian noise signal has a cusp at the origin and that this cusp is sharper when the average number, N0, of zero crossings in unit time is larger. When an underlying noise signal s(t) has a Gaussian distribution, the average number, N0, of zero crossings in unit time can be determined from Rice's formula:N0=Bs/πwhere Bs is the angular root-mean-square (rms) bandwidth (measured in radians per second) of signal s(t). Consequently, when a Gaussian noise signal s(t) is employed to generate a random binary waveform b(t), it is not possible to reduce the width of the correlation function Rb(τ) of the binary waveform by means other than the increase in the rms bandwidth Bs of the underlying noise signal s(t). Unfortunately, the generation of ultra wideband noise signals is very difficult in practice.